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18.03SC Practice Problems 24 Step and delta functions ⎧ 0 ⎪ ⎪ ⎨ 2t − 2 1. Let Q(t) = −1 2t ⎪ ⎪ ⎩ 5 for t < 1 for 1 < t < 2 for 2 < t < 3 for 3 < t (a) Sketch a graph of this function. Is it piecewise smooth? (b) Find the generalized derivative q(t) = Q� (t), and sketch it. (c) Describe a scenario which might be modeled by the equation ẋ + kx = q(t) (your choice of k) with rest initial conditions. (d) Describe a scenario which might be modeled by the equation 2ẍ + 4ẋ + 4x = q(t) with rest initial conditions. MIT OpenCourseWare http://ocw.mit.edu 18.03SC Differential Equations�� Fall 2011 �� For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.